Order-theoretic Invariants in Set-theoretic Topology

We present several results related to van Douwen’s Problem, which asks whether there is homogeneous compactum with cellularity exceeding c, the cardinality of the reals. For example, just as all known homogeneous compacta have cellularity at most c, they satisfy similar upper bounds in terms of Peregudov’s Noetherian type and related cardinal functions defined by order-theoretic base properties. Also, assuming GCH, every point in a homogeneous compactum X has a local base in which every element has fewer supersets than the cellularity of X. Our primary technique is the analysis of order-theoretic base properties. This analysis yields many results of independent interest beyond the study of homogeneous compacta, including many independence results about the Noetherian type of the Stone-Čech remainder of the natural numbers. For example, the Noetherian type of this space is at least the splitting number, but it can consistenly be less than the additivity of the meager ideal, strictly between the unbounding number and the dominating number, equal to c and greater than the dominating number, or equal to the successor of c. We also prove several consistency results about Tukey classes of ultrafilters on the natural numbers ordered by almost containment. We also characterize the spectrum of Noetherian types of ordered compacta and mostly characterize the spectrum of Noetherian types of dyadic compacta. Also, we show that if every point in a compactum has a well-quasiordered local base, then some point has a countable local π-base. Our secondary technique is an amalgam, a new quotient space construction that allows us to transform any homogeneous compactum into a path connected homogeneous